Network equilibrium with combined modes: models and solution algorithms

Transportation Research Part B 39 (2005) 223–254www.elsevier.com/locate/trb

Network equilibrium with combined modes:models and solution algorithms

Ricardo Garc�ıa a,*, Angel Mar�ın b

a Universidad de Castilla-La Mancha, E.U. Polit�ecnica de Almad�en, Plaza Manuel Meca, 1, 13.400-Almad�en,Ciudad Real, Spain

b Universidad Polit�ecnica de Madrid, E.T.S.I. Aeron�auticos, Plaza Cardenal Cisneros, 3, 28.040 Madrid, Spain

Received 2 September 2001; received in revised form 22 March 2003; accepted 12 May 2003

Abstract

In this paper we propose a new model for the equilibrium multi-modal assignment problem with

combined modes (MAPCM) for the case of asymmetric costs. MAPCM is stated on a generic passenger

assignment equilibrium model, on a generalized traffic assignment model, and on a nested logit distribution

as demand model which explicitly takes into account the choice of mode of transport and transfer node

among modal networks. This model is formulated as a variational inequality problem in the space of the

hyperpath flows and then solved by the disaggregate simplicial decomposition (DSD) algorithm.Illustrations of the model and of the numerical approach are reported on two test networks with

asymmetric cost functions.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Multi-modal network equilibrium models with combined modes; Urban transport management; Simplicial

decomposition; Variational inequalities

1. Introduction

The modeling of urban networks has witnessed a growing amount of research attention recentlyas the interest in advanced congestion management programs increases. Many of these models areformulated as mathematical programs with equilibrium constraints (MPEC). At the upper level

* Corresponding author. Tel.: +34-92-626-4007; fax: +34-91-336-6324.

E-mail addresses: [email protected] (R. Garc�ıa), [email protected] (A. Mar�ın).

0191-2615/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trb.2003.05.002

mail to: [email protected]

224 R. Garc�ıa, A. Mar�ın / Transportation Research Part B 39 (2005) 223–254

the central authority (leader) takes design decisions seeking to improve the network’s perfor-mance, but at the lower level it takes into account the user behavior.

The lower-level problem of the MPEC is a user’s behavioral model which evaluates the reac-tions of the users to the design network. In the strategic management of urban transportationsystems the so-called mode combined models are a proper tool in order to describe the future use ofthe transportation system. At present, single mode models have been much advanced; optimi-zation approaches, variational inequality approaches and stochastic equilibrium approaches havebeen developed and excellent results were obtained in applications (Evans, 1976; Florian et al.,1977; Erlander, 1977; LeBlanc and Farhangiam, 1981; Ludgren and Patriksson, 1997). Reviews ofthe state of the art of these models are given in Boyce (1984) and Fern�andez and Friesz (1983).

Important advances have been realized over the past 20 years in the formulation and analysis ofmulti-modal network equilibrium models (Florian, 1977; Florian and Spiess, 1983; Nagurney,1984; Wong, 1998; Ferrari, 1999). These models consider several alternatives to travel from anorigin to a destination by using a ‘‘pure’’ mode of transport such as the private car mode or thepublic transit mode. For example, assuming a logit type mode choice function which gives theproportion of trips taken on each mode according to generalized cost in the considered alter-natives.

Lam and Huang (1992) propose a combined trip distribution and assignment model for mul-tiple user classes in (multi-modal) road networks in which the link travel time is similar for alltraffic. Toint and Wynter (1995) dealt with the formulation of the asymmetric multi-class trafficassignment problem. They proposed a general functional form in order to avoid behavioralinconsistencies.

Bifulco (1993) proposes a stochastic multi-user equilibrium model in order to evaluate parkingpolices at urban central areas. This model consists of a Probit model as a demand model which isapplied to parking and path levels of choices, a road network model as a supply model, and asupply/demand interaction model. The extension of this model on the mode choice may be ef-fected using a hyper-network approach.

In urban transportation many trips use more than one mode of transport, such as the so calledpark’n ride, where the first part of the trip is taken by private car; then the trip is completed bytaking one or more public transit modes and by walking to the final destination. The promotion ofcombined trips requires adequate tools, which takes into account their attractiveness, will dependon road congestion, service frequency and fares (mode and parking), and all of these are, ingeneral, mutually related. But there are also two models that consider explicitly the combinedtrips. Florian and Los (1978) developed a model for parking lot allocation for park’n ride trav-elers. This model determines the intermediate origin–destination matrix for the first component ofthe combined trip, i.e., from home to the parking lot. The interest of this problem is to predictchanges in traffic flows that would result from changes in parking policy, such as decreases orincreases in parking lot capacities, additional parking facilities and changes in pricing of parkingfacilities. The drawback of this model is that the origin–destination (O–D) matrix of combinedtrips is an input, and it does not consider the elastic nature of this type of demand in function ofthe facilities of transferring.

The choice of the transfer node (interchange) in the above models is a direct consequence of thepaths that are generated during the computation of the equilibrium in the multi-modal networkand the assignment of the resulting O–D trip matrix to the corresponding combined mode paths.

R. Garc�ıa, A. Mar�ın / Transportation Research Part B 39 (2005) 223–254 225

In most large urban areas, planning policies sustain the use of public transport and try to reducethe amount of personal vehicle traffic through the design of interconnecting high-quality publictransport systems such the metro and rapid regional train lines. This can be achieved through theintroduction of urban multi-modal interchanges where it is possible to change to a different modeof transport. A proper approach to the modeling of this problem must explicitly take into accountthe choice of urban multi-modal interchange. In this context the access to these nodes is taken, forexample, by car or bus feeder lines, so the total travel cost via two different interchanges may notbe meaningfully distinct, and it does not determine the user’s choice, since both interchanges maybe used by the trip-makers depending on relative attractiveness of transfer nodes, due to factorsnot included in the user’s generalized cost perception such as: security, protection, comfort, etc.This problem requires a modeling of the choice of the transfer network on the public transportnetwork and a passenger equilibrium assignment model for this network.

Network equilibrium models which explicitly identify the choice of transfer node made by thetravelers who undertakes a combined mode trip have not received much attention in literature.Fern�andez et al. (1994) presented several approaches to formulating network equilibrium modelswith combined modes. Their model (P3) explicitly includes the choice of the transfer point(parking lot) for the combined mode by means of a nested logit model. This model is formulatedassuming symmetric cost functions. This limitation reduces the range of applicability of themodel. Garc�ıa and Mar�ın (2002) formulate a network design problem using P3 in order to designparking facilities in park’n ride trips and Garc�ıa and Mar�ın (2001) propose a mathematical modelbased on P3 for the location of urban multi-modal interchanges.

Our approach to the formulation of network equilibrium models with combined trips may beconsidered as an extension of the model of Fern�andez et al. (1994) to asymmetric cost functions.The transit networks with overlapping and competitive lines, model user route choice strategiesthrough the hyperpath concept introduced by Nguyen and Pallotino (1988). To deal with thisproblem it is necessary to include non-symmetric travel costs and a variational inequality ap-proach. In literature several transit equilibrium assignment models with these characteristics existsuch as Nguyen and Pallotino (1988), Spiess and Florian (1989), Cea and Fern�andez (1993), Wuet al. (1994), etc.

In our model both the transit assignment problem and the asymmetric traffic assignmentproblems are considered. This model works on a generic transit equilibrium assignment modelformulated in the space of hyperpath flows and on an asymmetric road network. Theseassumptions give a lot of flexibility to the model in order to consider public transport assignmentmodels or the representation of the road network. We assume a deterministic user principle togovern the choice of the route in each modal network, but we also discuss the means of extendingthe model to stochastic assignment framework.

The proposed model employs a nested logit model to describe the user’s choice of mode andtransfer node. To deal with other demand models the fixed-point approach of Cantarella(1997) could be used. He presents a general fixed-point model for multi-mode multi-user deter-ministic and/or stochastic equilibrium assignment with elastic demand. He extended andgeneralized existing models in several ways. Users of different classes may have different beha-vioral characteristics as well as sets of available routes and modes. In the general point-fixedapproach of Cantarella (1997) the two algorithms are elaborated to the general case of theproblem.


There is a considerable literature on the topic of algorithms for general traffic equilibria. See forexample Patriksson (1994). Decomposition algorithms of the nonlinear Jacobi type, along withprojection algorithms are the predominant algorithms proposed.

The column generation/simplicial decomposition strategy for the (asymmetric) traffic assign-ment problem has been utilized in different forms by Bertsekas and Gafni (1982), Pang and Yu(1984), Lawphongpanich and Hearn (1984), Marcotte and Gu�elat (1988), Larsson and Patriksson(1992), Montero and Barcel�o (1996), Larsson et al. (1997). In Wu and Florian (1993) an adap-tation of the simplicial decomposition algorithm for the transit equilibrium assignment problemwas proposed. However, few references in literature exists on the use of the simplicial decom-position algorithms for multi-modal network equilibrium models. The majority of these appli-cations are used in separable models. In Garc�ıa et al. (submitted for publication) the model P3 issolved using a class of nonlinear column generation/simplicial algorithms given in Garc�ıa et al.(2003).

The means of representation of the feasible set of the restricted master problems can be used torealize a taxonomy of the class of simplicial decomposition algorithms. The so-called aggregatedversions use link flow pattern and the disaggregated versions use route flow pattern. This secondapproach was originated in Larsson and Patriksson (1992) whose development the of disaggregatesimplicial decomposition (DSD) for the symmetric traffic assignment problem was formulated asan optimization problem.

This paper presents a disaggregated simplicial decomposition algorithm for the variationalinequality formulation of the equilibrium multi-modal assignment problem with combinedmodes, for short MAPCM. It alternates between the generating of the multi-modal shortesthyperpath and the approximate solving of a disaggregate master variational inequality subject tosimple linear constraints. The main characteristics of this approach are that it uses route flows andcan utilize any valid technique for approximately solving the disaggregate master problem.

The main motive for considering a disaggregate version of the simplicial scheme instead of anaggregate one is that MAPCM is formulated in the space of hyperpath flow. It allows us to solvethe more general formulation of MAPCM because some passenger transit equilibrium assignmentmodels cannot be transformed into a variational problem in arc space. This is in contrast to thetraffic assignment network equilibrium where the variational inequality problem defined in thespace of arc flows and the transformed one, in the space of path flows, are equivalent. Anothermotivation for considering this algorithm is its excellent restarting capabilities making the ap-proach profitable for more complex models, where MAPCM arise as subproblems.

The rest of the paper is organized as follows. The model formulation is presented in Section 2.In Section 3, we propose a simplicial decomposition algorithm. Computational results are re-ported in Section 4. Finally, conclusions are established in Section 4.2.

2. Network equilibrium model with combined modes

The two main modeling issues which arise when combined mode trips occur are modeling thechoice of the combined trip, and modeling the choice of transfer node. As a consequence, it isimportant to decide which choices are modeled by mode choice (demand) and which by routechoice (network). The two main components of the model are transportation demand and the


transportation network. The demand model is taken from Fern�andez et al. (1994) and it is brieflyreviewed in Section 2.1. The transportation network model is discussed in Section 2.2. Theobjective of Sections 2.3 and 2.4 is to show the mechanisms to formulate the equilibrium con-ditions as a variational inequality problem in the space of hyperpath flows. In Section 2.5 themodel proposed is extended to deal with stochastic assignment approaches and with elastic de-mand.

2.1. Modeling of the demand

To elaborate a planning tool in order to promote the use of combined trips such as the park’nride, the models should take into account the competition among the main modes of transport(see Ben-Akiva and Bowman, 1998). We have dealt with the following modes: car, transit withwalk access, and transit with auto access.

It is assumed that an origin–destination matrix, f�gxgx2W of trips by all modes is known andfixed for our planning period. The total demand for the O–D pair x is disaggregated by thefollowing transportation alternatives:

(a) Car. The number of users that take their trips by car is denoted as gax.(b) Transit with walk access or bicycle access. The number of passengers using transit with walk

or bicycle access is denoted as gbx.(c) Transit with auto access (park’n ride). The number of users that take their trips by the car-

transit alternative for the O–D pair x is denoted as gcx, and the amount of them transferringat transfer node t is denoted as gcx;t

The mode of transport (c) is a combined mode trip over the urban/suburban transportationsystem. This includes the auto-regional rail, auto-metro, auto-regional bus, etc. alternatives. Themode of transport (b) includes trips such as bus-metro as well as the pure modes: metro, bus,regional rail, etc. In this paper, for clarity of the exposition, we consider that only the users of thepark’n ride mode choose the transfer node explicitly (modeled by the demand model), but thischoice is determined solely on route choice considerations for the users’ transit trip. In othercontexts a user’s choice of the transfer node among transit networks such as regional rail andmetro networks may be taken into account.

We consider a nested logit distribution to model the disaggregation of the demand by modesand interchanges.

The proportion of trips is given for each origin–destination pair x, and each mode k 2 fa; b; cg,by the formula

GkxðU �

xÞ ¼expf�ðak þ b1U

k�x ÞgP

k02fa;b;cg expf�ðak0 þ b1Uk0�x Þg ; ð1Þ

where Uk�x is the user’s perception of the generalized cost of traveling of the pair x by mode k, that

corresponds to a user optimal route choice of the network, fU �xg is the vector of generalized costs

for all the modes present, and ak, b1 are parameters of the logit model.


The logit and nested logit models are widely used in literature (see Hunt and Teply, 1993;Fern�andez et al., 1994) to model the choice of parking. The model explicitly includes the choice oftransfer node t (parking) in the context of the park’n ride mode, (c), and demand x by means of alogit function such as

Gcx;tðUc�

x Þ ¼expf�ðact þ b2U

c�x;tÞgP

t02Tx expf�ðact0 þ b2Uc�x;tÞg

; ð2Þ

where Tx is the available set of transfer nodes for the pair x, and park’n ride alternative.The utility for the alternative (c) of the pair x, Uc�

x for use in Eq. (2), is computed as a ‘‘log-sum’’ of the utilities through each interchange, i.e.,

Uc�x ¼ �1

b2

lnXt2Tx

expf

� ðact þ b2Uc�x;tÞg

!: ð3Þ

The parameters act represent the disutility of transfer node t, due to factors not included in theuser’s generalized cost perception Uc�

x;t such as: security, protection, comfort, ticketing, handling ofbaggage, etc., and b2 ponders the importance of the generalized cost perception in the transfernode choice decision process.

The relationships (1) and (2) define the disaggregation of the demand. The number of userswhose travel by mode k 2 fa; b; cg for the demand pair x is computed as

gkx ¼ GkxðU �

xÞ�gx; ð4Þ

and the number of trips of the demand x using park’n ride and transferring through the inter-change t, that is denoted as gcx;t is given by

gcx;t ¼ GcxðU �

xÞGcx;tðUc�

x Þ�gx: ð5Þ

2.2. Modeling the transportation network

In this paper we consider a multi-modal transportation network G ¼ ðN;LÞ which is formedby a classical road network Ga ¼ ðNa;AÞ, a public transportation network Gb ¼ ðNb;BÞ, andtransfer nodes between both, called urban multi-modal interchanges.

The demand model considers a set of O–D pairs x ¼ ði; jÞ, where i is an origin and j a desti-nation, denoted as W . A combined trip to satisfy an O–D pair x ¼ ði; jÞ is composed of the tripfrom the origin i to parking lot t, and from the interchange t to final destination j in the publictransportation system. This type of trips adds a new set of O–D pairs of share demand in eachtransportation network Ga and Gb. We define

W a ¼ W [ fði; tÞ j t 2 Tx; and x ¼ ði; jÞ 2 W g;W b ¼ W [ fðt; jÞ j t 2 Tx; and x ¼ ði; jÞ 2 W g;

where Tx is the set of interchanges used by the pair x.We assume that the road path cost mapping Cað�Þ is known and it is available to compute their

components CpðhÞ for all road path p (p 2 Pa) and feasible flow vector h on Ga.


The public transportation network may consist of a set of lines of subway, bus and regional raillines, stops and walking links. This network is abstracted into a graph model, which contains fourtypes of arcs: walk arcs, wait arcs, in vehicle arcs and transfer/alight arcs. To be able to include theproblem of existing common lines of bus, subway or regional rail it is necessary to use the conceptof strategy. This leads us to use the hyperpath framework that is more realistic than the concept ofpath.

Over the past four decades, a variety of passenger assignmentmodels in congested transit networkhave been proposed. These models are based on different hypotheses in order to model passengertravel behavior (see Bouza€ıene-Ayari et al., 1998). In this paper we do not assume a specific pas-senger assignment model of the literature, because the focus of the paper is the integration betweenroad network and public transportation network and the user’s choice of transfer node.

In this paper it is only required that it is possible to compute the hyperpath cost CpðhÞ for allhyperpath p and for all feasible hyperpath flow vector h on the network Gbðp 2 PbÞ. These (ex-pected) hyperpath costs may be computed in terms of travel costs, waiting times and distributionprobabilities between attractive lines at the stops.

The multi-modal network G integrates both transportation networks Ga and Gb, and a user cantravel on G using a path p 2 Pa, a hyperpath p 2 Pb or a combined hyperpath p ¼ ðpa; pbÞ wherepa 2 Pa and pb 2 Pb. We denote Pc as the set of combined hyperpaths, that is Pc � Pa � Pb andP ¼ Pa [ Pb [ Pc as the set of path/hyperpath in multi-modal transportation network G. We de-note as Px the subset of hyperpaths of P so that the O–D pair x is satisfied.

There are four main issues in the modeling of combined hyperpaths:

(1) Compatibility of flows. The first task of modeling is derived from the fact that in road net-works the flows are computed in vehicular units. On the other hand, the flow are computedin passenger units on public transportation networks. The car occupancy rate for each de-mand can be introduced in order to transform vehicular units into passenger units. We denotethis parameter as cx with x 2 W . We assume that the hyperpath flow hp for all p 2 P and linkflow in Gb are valued in passenger units, and the road link flows are valued in vehicular units.

To relate the flow in the paths with the flow in the links we consider the constraints:

fl ¼Xx2W

1

cx

Xp2Px

dlphp

!; l 2 A; ð6Þ

fl ¼Xx2W

Xp2Px

dlphp

!; l 2 B: ð7Þ

These constraints can be expressed in matrix form as f ¼ df h, where df is the link/pathincidence matrix in the multi-modal network G whose elements dlp are defined by

dlp ¼1cx; if link l 2 A and l is used by the path p 2 Px;

1; if link l 2 B and l is used by the path p 2 Px;0; otherwise:

8<: l 2 L; p 2 Px; x 2 W :

(2) Compatibility of cost units. Note that the road path cost is expressed in car cost units. On theother hand, the hyperpath cost in Gb is expressed in passenger cost units. To homogenize these


two costs it is necessary to use the car occupancy rates in order to share the generalized carcost by the passengers in the same car.

The individual passenger cost on the road network can be computed using expressions ofthe type

CpðhÞ ¼ tpðhÞ þtpðhÞcx

;

where tpðhÞ represents the time cost and tpðhÞ takes into account the consumption of fuel, theparking fares, and other attributes of the generalized cost that can be shared by the passengersin the same car.

Note that the same path in the car network which joins an origin i and a transfer node t,leads to different costs depending on which combined path is integrated. This is due to thedifferent car occupancy rates for different O–D pairs x. To make this fact evident we intro-duce the notation Cpa;xðhÞ, pa 2 Pa

x, x 2 W a.(3) Compatibility of perceptions of the travel cost. An important question is how the generalized

cost should be specified in the demand model in order to ensure compatibility with the corre-sponding measures obtained from the network model. The travel costs in the public transpor-tation network and road network have a different nature. For instance, it may be judged thatwalking times are far more important than in-car travel times. The model includes the param-eters ha and hb to homogenize the costs in both networks.

We define

CpðhÞ ¼haCpðhÞ; p 2 Pa

x; x 2 W ;hbCpðhÞ; p 2 Pb

x; x 2 W ;

�ð8Þ

which represents the path/hyperpath costs on each network Ga and Gb.The combined hyperpath cost of a hyperpath p ¼ ðpa; pbÞ 2 Pc

x is computed as the sum ofthe cost of each component

CpðhÞ ¼ haCpa;xðhÞ þ hbCpbðhÞ: ð9Þ

(4) The modeling of transfer flows. The model includes transfer links to represent parking lots.The parking time, walking time to the station, parking fares, parking capacity may be in-cluded in the cost functions. Some of these parking characteristics are time-dependent butthe description of flow pattern is based on a steady state quantity. It is necessary to effect somesimplifications for these measures to be compatible.

The parking capacity is computed as a mean value of the available parking places for theplanning period considered. It is assumed that the parking fares are independent of theparking duration. This may be considered unrealistic for parking lots located downtown butnot for suburban areas which have the goal to promote commuting.

2.3. Equilibrium conditions (MAPCM)

The equilibrium conditions, MAPCM, are given by the intersection of the following threesubsets of conditions:


C1: The choice of route. A network equilibrium model, in which one aims to provide a macro-scopic description or prediction of the traffic-passenger volumes resulting from route choicemade in the multi-modal network, must therefore be based on a route choice behavioral prin-ciple. The MAPCM assume that Wardrop’s user-optimal principle governs the route choicein every subnetwork Ga and Gb.

It can be formulated as

C�p � Uk�

x

¼ 0; if h�p > 0;

P 0; if h�p ¼ 0;

(p 2 Pk

x; x 2 W ; k 2 fa; bg;

C�p � Ua�

it;x

¼ 0; if h�p > 0;

P 0; if h�p ¼ 0;

(p 2 Pa

it ; ði; tÞ 2 W a;

C�p � Ub�

tj

¼ 0; if h�p > 0;

P 0; if h�p ¼ 0;

(p 2 Pb

tj; ðt; jÞ 2 W b;

ð10Þ

where, Pcx;t is the subset of hyperpaths of P

cx which go through the transfer node t. Pa

it is the setof paths connecting the origin i with the transfer node t on the network Ga. Pb

tj is the set ofhyperpaths connecting transfer node t with destination j on the network Gb.The user’s perception of the generalized cost of a path for park’n ride trips at equilibrium is

Uc�x;t ¼ Ua�

it;x þ Ub�tj;x; t 2 Tx; x ¼ ði; jÞ 2 W : ð11Þ

C2: The choice of mode of transport. The proportion of users in every mode k 2 fa; b; cg, foreach pair x 2 W , is given as demand function Gk

x defined as (1), where the utility for the alter-native (c), Uc�

x , is computed as (3). When these proportions are achieved no user has theincentive to change the mode unilaterally.

C3: The choice of the transfer node. The proportion of park’n ride mode users that choose eachtransfer point t 2 Tx, for each pair x 2 W is given by the demand function Gc

x;t defined by (2).When these proportions are achieved a none user has the incentive unilaterally change thetransfer point chosen.

2.3.1. Unified equilibrium conditionsIn the previous subsection the equilibrium conditions have been introduced by means of three

different sets of conditions C1, C2 and C3. The first for the choice of the route/strategy in eachmodal network, the second for the choice of mode of transport and the third for the choice oftransfer node on park’n ride trips. In this subsection we give a unified formulation of the equi-librium conditions. If a user chooses a path on the multi-modal network G, he/she has implicitlychosen a mode of transport, a route/strategy and a transfer node for the park’n ride alternative.We consider that each hyperpath on the multi-modal network has an extended equilibrium costwhich models these implicit choices. The users’ behavior is modeled by a version of Wardrop’suser-optimal principle in which the extended cost of the journey on all the hyperpaths actuallyused are equal, and less or equal to those which would be experienced by a single user on anyunused hyperpath.


We denote as X the set of feasible hyperpath flow on multi-modal network G defined as

X ¼ h 2 RMX

k2fa;b;cg

Xp2Pk

x

hp

��8<: ¼ �gx for all x 2 W ; hP 0

9=;; ð12Þ

where M is the cardinality of the set P .Now we relate the share demand variable g with the feasible hyperpath flow. The first con-

straints are the supply of demand, and the modal split constraints which produce the partitioningof the demand by modes of transport. The total demand for each O–D pair x must be the addingof the demand in each alternative

�gx ¼X

k2fa;b;cggkx; ð13Þ

where the demand for combined mode (c) in O–D pair x is given by

gcx ¼Xt2Tx

gcx;t ð14Þ

and the relationships between the share demand variables with the flow paths are stated as

gkx ¼Xp2Pk

x

hp; k 2 fa; b; cg; x 2 W ; ð15Þ

gcx;t ¼Xp2Pc

x;t

hp; t 2 Tx; x 2 W : ð16Þ

The relationships (15) and (16) can be expressed in matrix form as

g ¼ dgh; ð17Þ
where dg is the share demand/path incidence matrix.
To formulate the equilibrium conditions we consider the inverse of the demand model kpðgÞwhich is defined as result of Theorem 2.1. Using the constraints (17)

KðhÞ ¼ kðdghÞ;
where kð�Þ ¼ ðkpð�ÞÞp2P . This term takes into account the user’s choice of mode and transfer node.The extended cost is the adding of the transportation cost to the cost associated to the choice ofthe mode of transport and of the interchange.
Theorem 2.1 gives the rule for the computing of these extended costs and it characterizes thepaths used at the equilibrium to satisfy a pair of demand x. The coefficient k�x plays the role of theextended cost in the equilibrium for the demand x.

Theorem 2.1 (Unified equilibrium condition of MAPCM). A vector h� 2 X is an equilibriumpath flow vector for the MAPCM if and only if there exists the set of values k�x, for all x 2 W sothat

½Cpðh�Þ � Kpðh�Þ� � k�x¼ 0; if h�p > 0;P 0; if h�p ¼ 0;

�for all p 2 Px; for all x 2 W ; ð18Þ


where KpðhÞ ¼ kpðdghÞ and kpð�Þ is defined by

kpðgÞ ¼� ln gkx þ ak

b1; if p 2 Pk

x; k 2 fa; bg; x 2 W ;

� ln gcx þ ak

b1�� ln gcx þ ln gcx;t þ act

b2; if p 2 Pc

x;t; w 2 W :

8><>: ð19Þ

Proof. Now, we prove that if h� satisfies the conditions (18) then h� satisfies the equilibriumconditions C1, C2 and C3.

First, we prove the condition C1. Consider the partition fPax; P

bx; P

cx;tg of the set Px. Let p1; p2 be

two paths with positive flow, belonging to the same component of the partition. If k 2 fa; bg andusing (18), it is held

k�x ¼ Cp1ðh�Þ � Kp1ðh�Þ ¼ Cp2ðh�Þ � Kp2ðh�Þ:

The relationship (19) implies that �Kpðh�Þ is constant on the paths of the same component Pkx

where k 2 fa; bg. We denote this value as ekx which depends only on the demand variable gkx, andekx ¼ �Kp1ðh�Þ ¼ �Kp2ðh�Þ.

The extended cost is the sum of a cost which depends on the path pi plus the value ekx, that onlydepends on the mode of transport used for the O–D demand pair x. Thus we obtain thatCp1ðh�Þ ¼ Cp2ðh�Þ. This implies that all the hyperpaths used to satisfy the same pair of demand xbelonging to the same component have the same cost. We denote as

Uk�x ¼ Cpðh�Þ; if p 2 Pk

x; k 2 fa; bg and h�p > 0: ð20Þ
It is held by (18) that k�x 6Cp þ ekx, and thus
Uk�x ¼ k�x � ekx 6Cp þ ekx � ekx ¼ Cp; for any path p 2 Pk

x; k 2 fa; bg: ð21Þ
The relationships (20) and (21) show that the hyperpaths of the pure modes (a) and (b) satisfy
the user-equilibrium conditions.Now we shall consider the case p1; p2 2 Pc

x;t. We first assume that p1 and p2 are two combinedhyperpaths with positive flow. Analogously to the above discussion, it is held that

k�x ¼ Cp1ðh�Þ þ ecx;t ¼ Cp2ðh�Þ þ ecx;t;

where ecx;t depends on gcx and gcx;t, and this value is the same for any hyperpath of Pcx;t. This implies

that Cp1ðh�Þ ¼ Cp2ðh�Þ.We denote

Uc�x;t ¼ Cpðh�Þ; if p 2 Pc

x;t and h�p > 0: ð22Þ

It holds by (18)

Uc�x;t ¼ k�x � ecx;t 6Cp þ ecx;t � ecx;t ¼ Cp for any path p 2 Pc

x;t: ð23Þ

Consider a hyperpath p0 ¼ ðp0a; p0bÞ 2 Pcx;t with positive flow. The hyperpath cost of p0 can be

expressed as

Cp0 ¼ Cp0a;xðh�Þ þ Cp0b

ðh�Þ:


The relationships (22) and (23) show that p0 is the shortest hyperpath from i to j through theinterchange t. Using Bellman’s optimality principle the path p0a, and the hyperpath p0b must also beoptima. This means that the minimum cost from i to t, that we denote as Ua�

it;x, is Cp0a;xðh�Þ andfrom t to j, that we denote as Ub�

tj , is Cp0bðh�Þ. We obtain

Ua�it;x ¼ Cp0a;xðh

�Þ; Ub�tj ¼ Cp0b

ðh�Þ:

On the other hand, let p ¼ ðpa; pbÞ 2 Pcx;t. Using the optimality of Ua�

it;x and Ub�tj , we obtain

Ua�it;x 6Cpa;xðh�Þ; Ub�

tj 6Cpbðh�Þ:

This completes the proof of C1.Now we show the condition C3. Consider that g ¼ dgh�. Let p 2 Pc

x;t so h�p > 0, then Cp ¼ Uc�x;t

and using the expressions (18) and (19), we obtain the relationship

k�x ¼ln gcx;t þ act

b2

þ kcx þ Uc�x;t; ð24Þ

where

kcx ¼ ln gcx þ ac

b1

� ln gcxb2

; ð25Þ

and removing gcx;t from (24) we obtain

gcx;t ¼ expð�fact þ b2Uc�x;tgÞ expðb2

~kcxÞ; ð26Þ

where ~kcx ¼ k�x � kcx.Using (14) we obtain

gcx ¼Xt2Tx

expð�fact þ b2Uc�x;tgÞ expðb2

~kcxÞ; ð27Þ

and removing ~kcx from (27) we obtain

~kcx ¼ 1

b2

ln gcx þ Uc�x ; ð28Þ

where Uc�x is the ‘‘log-sum’’ of the utilities, see (3).

Substituting the expression of ~kcx in (26) we obtain

gcx;t ¼exp�ðact þ b2U

c�x;tÞP

t02Tx exp�ðact0 þ b2Uc�x;t0 Þ

gcx:

The previous condition shows that the condition C3 is held.Now we prove that the condition C2 is held. Substituting the value of kcx given in (25) and the

value of ~kc given in (28) in the relationship k�x ¼ kcx þ ~kcx, we obtain

k�x ¼ kcx þ 1

b2

ln gcx þ Uc�x ¼ ln gcx þ ac

b1

� 1

b2

ln gcx þ 1

b2

ln gcx þ Uc�x ¼ ln gcx þ ac

b1

þ Uc�x : ð29Þ


On the other hand, using the equilibrium condition (18) we obtain

k�x ¼ ln gkx þ ak

b1

þ Uk�x ; k 2 fa; bg; ð30Þ

and removing gkx from (29) and (30), we obtain

gkx ¼ expð�fak þ b1Uk�x gÞ expðb1k

�xÞ; k 2 fa; b; cg: ð31Þ

Using the relationships (13) and (31) to remove the value of k�x, we obtain

k�x ¼ �1

b1

ln�gxP

k2fa;b;cg expð�fak þ b1Uk�x gÞ

" #: ð32Þ

Substituting (32) in (31) it is shown that the condition C2 is held.The other implication is that if h� satisfies C1, C2 and C3 then hold (18), is shown using similar

arguments. The proof of the ‘‘only if’’ part is obtained following the proof of the ‘‘if’’ part in acontrary direction, and for this reason it is omitted. h

2.4. Mathematical formulation of the equilibrium conditions

Let C � K : RM 7!RM be the vector valued function whose components are CpðhÞ � KpðhÞ for allp 2 Px, and for all x 2 W . We have the following theorem.

Theorem 2.2 (Inequalities variational formulation for MAPCM). A vector h� 2 X is an equilib-rium path flow vector for MAPCM if and only if it satisfies the following variational inequalityproblem.

[TAP-MVIPðX;C � KÞ]: Find h� 2 X so that

½Cðh�Þ � Kðh�Þ�Tðh� h�ÞP 0 for all h 2 X:

Proof. This proof is based on the one given in Ferrari (1999). It is easy to verify that h� is asolution of VIPðx;C � KÞ if and only if it is a solution of the following optimization problem:

minf/ðhÞ : h 2 Xg where /ðhÞ ¼ ½Cðh�Þ � Kðh�Þ�Tðh� h�Þ: ð33Þ
We write the constraints which define the set X as follows:
GxðhÞ ¼X

k2fa;b;cg

Xp2Pk

x

hp � �gx ¼ 0; for all x 2 W ; ð34Þ

spðhÞ ¼ �hp 6 0; for all p 2 P : ð35Þ
Since /ðhÞ and constraints (34) and (35) are linear, h� is a solution of problem (33) if it verifies thefollowing KKT conditions:
Cðh�Þ � Kðh�Þ þXp2P

uprspðh�Þ þXw2W

qxrGxðh�Þ ¼ 0;

where the multipliers up, for all p 2 P are nonnegative and verify the complementary slackness (CS)condition, upspðh�Þ ¼ 0, whereas multipliers qx for all w 2 W may have any sign. If h�p > 0 with


p 2 P , from the CS condition up ¼ 0 and Cpðh�Þ � Kpðh�Þ ¼ �qx; on the other hand, if h�p ¼ 0, p 2P we have up P 0 and thus Cpðh�Þ � Kpðh�ÞP � qx.

Letting k�x ¼ �qx, using Theorem 2.1 we obtain that h� is an equilibrium vector if and only if itis a solution of VIPðX;C � KÞ. h

2.5. Extensions of MAPCM

In this subsection we show some possible ways to extend MAPCM. The first one is a stochasticassignment model and the second one deals with the elastic demand case. We show only themodified extended costs necessary to formulate the new models as a variational inequalityproblem. The proof of the equivalence between equilibrium conditions and the variationalinequality problem can be derived by using the same arguments as the ones given in Sections 2.3and 2.4.

The condition C1 is a version of the deterministic user equilibrium principle (DUE). It ispossible to modify C1 in order to consider stochastic versions of MAPCM where the users makesystematic errors in their perception of the travel cost. For example, an interesting case of sto-chastic MAPCM occurs when the path probabilities are given by a logit function:

PrðpÞ ¼ expð�b3CpðhÞÞPp02P expð�b3C

0pðhÞÞ

; p 2 P; P 2 fPkx; P

bt;j; P

ai;tg: ð36Þ

In this particular case, it is easy to show that the new equilibrium conditions are equivalent to thevariational inequality problem which is obtained by the replacing the path/hyperpath cost CpðhÞby CpðhÞ þ 1

b3ðlnðhpÞ þ 1Þ.

In this model all paths of the same component P have the same cost as the equilibrium and thisis Cpðh�Þ þ 1

b3ðlnðh�pÞ þ 1Þ where h� is the equilibrium path flow vector. These values will be used in

the demand model to effect the disaggregation by transport modes and transfer nodes. In anotherway, the condition (36) may be introduced in the nested logit model as a new nest which representsthe choice of route according to the formula (36). In order to fit the nested logit model with thesequence of choices taken by an user, the first nest is associated with the mode of transport, thesecond nest with the transfer node and finally the third nest with the route. In this case, the ex-tended cost associated with demand model is defined as

kpðgÞ ¼

� ln gkx þ ak

b1

�� ln gkx þ ln hp þ 1

b3

;

if p 2 Pkx; k 2 fa; bg; x 2 W ;

� ln gcx þ ak

b1

�� ln gcx þ ln gcx;t þ act

b2

�� ln gcx;t þ ln hp þ 1

b3

;

if p 2 Pcx;t; w 2 W :

8>>>>>>>>><>>>>>>>>>:

ð37Þ

The above two models satisfy Eq. (36) but in the second approach the path/hyperpath cost isspecified in the nested logit model using the ‘‘log-sum’’.

MAPCM is formulated as a problem with fixed demand, but it can be extended to the elasticdemand case, where the trip rates in the pair x are modeled as functions of a measure of the


accessibility between the origin and destination. Again, the ‘‘log-sum’’ of the mode travel costs isused to this objective and it is denoted as Ux;x 2 W .

The elastic demand case extends the traveler’s hierarchic choices given in Section 2.1 toincorporate the choice of making the trip. A traveler has an incentive to make a trip in function ofthe utility. After, he/she chooses the transport mode and finally chooses the transfer node.

To extend MAPCM to the case of elastic demands, let the demand for transportation for thepair x be a function of the Ux, x 2 W , i.e.,

�gx ¼ GxðUxÞ; x 2 W ;

where Gx is nonnegative, continuous and strictly decreasing in positive real numbers. Under theseassumptions, the demand function is invertible.

The new extended costs, denoted by KpðhÞ, related to the demand model are defined as

KpðhÞ ¼ KpðhÞ þ G�1x ð�gxÞ þ

ln �gxb1

; p 2 Px; x 2 W ;

where KpðhÞ is defined in Theorem 2.1. The characterization of the equilibrium conditions given inTheorem 2.1 is hold for the values k�x ¼ 0, x 2 W .

3. A column generation/simplicial decomposition algorithm

In this section we deal with an instance of the class of generation columns/simplicial decom-position (CG/SD) algorithms developed in Patriksson (1999) applied to the variational inequalityproblem VIPðX;C � KÞ.

These algorithms iterate by alternately solving two variational inequality subproblems: the so-called column generation problem (CGPVIP) and the so-called restricted master problem (RMP-VIP). CGPVIP generates a new column of the feasible region by means of the approximation ofthe cost mapping on the original feasible region, and RMPVIP solves the original cost mappingover a subset based on previously generated points (columns). RMPVIP is a variational inequalityproblem with simple constraints whose solution defines the next point to approximate the costmapping at CGPVIP. Formally, the previous iterative scheme is stated as follows.

(1) CGPVIPuð‘Þ. Let u be a function RM �RM 7!RM . At the iterate h‘�1 the CGPVIP is definedas [CGPVIPðX;uðh‘�1; �ÞÞ]: find �h‘ 2 X so that

uðh‘�1; �h‘ÞTðh� �h‘ÞP 0; for all h 2 X; ð38Þ

where uðh‘�1; �Þ is an approximation of the cost mapping ðC � KÞð�Þ at the point h‘�1. In thispaper we deal with the choice of the mapping uðh‘�1; sÞ ¼ Cðh‘�1Þ � Kðh‘�1Þ.

(2) RMPVIPð‘Þ. The original variational inequality problem is solved in a compact convex subsetof the feasible set, i.e. let X‘ be a convex and compact set such as X‘ � X which is defined bythe CG/SD algorithm, so the RMPVIP at the iteration ‘ is defined as [RMPVIPðX‘;C � KÞ]:Find h‘ 2 X‘ so that

½Cðh‘Þ � Kðh‘Þ�Tðh� h‘ÞP � �‘; for all h 2 X‘;

Table 1

The CG/SD algorithms

0. (Initialization): Choose an initial point h0 2 X, a tolerance parameter ��, a sequence of positive number so

that f�‘g ! 0, and let ‘ :¼ 1. Set X0 ¼ fh0g.1. (Column generation problem): Solve CGPVIPuð‘Þ. Let �h‘ be the solution.

2. (Termination criterion): If h‘�1 solves CGPVIPuð‘Þ then Stop (h‘�1 solves TAP-MVIPðX; �C � KÞ).Otherwise, continue.

3. (Convergence test): Let the merit function GAPuðh‘�1Þ defined as

GAPuðh‘�1Þ ¼ minh2X

uðh‘�1; hÞTðh‘�1 � hÞ ¼ uðh‘�1; �h‘ÞTðh‘�1 � �h‘Þ:

If GAPuðh‘Þ6�� then stop.

4. (Set augmentation): Let X‘ be a convex and compact subset of X so that �h‘ 2 X‘ � X‘�1.

5. (Restricted master problem): Find an �‘-solution h‘ of RMPVIP(‘).6. (Update): Let ‘ :¼ ‘þ 1 and return to Step 1.


where �‘ is a given positive number. The point h‘ is called an �‘-optimal solution of RMPVIPð‘Þand defines the next CGPVIP.

This algorithm belongs to the CG/SD class and may be stated as Table 1.

3.1. Solving the restricted master variational inequality problem

For a CG/SD algorithm to be operative two tasks must be discussed. The first one is thedefinition of the feasible region of RMPVIP, and the second one is the solution procedure forRMPVIP. These two problems are related, and they must be dealt with simultaneously. We beginwith the definition of X‘.

Table 2 summarises a solution approach for CGPVIPð‘Þ and it generates a hyperpath for O–Dpair x in each iteration (and it is denoted as p‘x). We consider a decomposition of the hyperpathflow �h‘ by O–D pairs, that is

�h‘ ¼ ð�h‘xÞx2W ;

where �h‘x is the hyperpath flows associated with the O–D pair x at the iteration ‘.In function of the current hyperpaths set for the O–D pair x, that is denoted as Q‘�1

x , andhyperpath flow set for x, that is denoted as P‘�1

x , we define for each pair x 2 W and for theiteration ‘

P‘x ¼ P‘�1

x [ f�h‘xg; Q‘x ¼ Q‘�1

x [ fp‘xg;

and the set X‘ can be defined as the following Cartesian product set

X‘ ¼Yx2W

X‘x; ð39Þ

where X‘x ¼ convðP‘

xÞ, where convðP‘xÞ is the convex hull of the points of P‘

x. CGPVIPuð‘Þassigns the total demand �gx to the hyperpath p‘x, then the columns of P‘

x are of the formð0; . . . ; �gx; . . . ; 0Þ, and the sets X‘

x, can be expressed as

Table 2

Solution approach for CGPVIP(‘)

1.0. (Initialization): Let h‘�1 2 X, compute f ‘�1 ¼ df h‘�1 where df is the matrix form of the equations given

in (13), g‘�1 ¼ dgh‘�1, and update the link cost mapping cðf ‘�1Þ.1.1. (Shortest path computation on Ga):

(a) For every x 2 W , based on the current cost caðf ‘�1Þ, compute a shortest path pax.Denote Ua�

x ¼ �Cpaxðh‘�1Þ.(b) For every ði; tÞ 2 W a � W , based on the current cost caðf ‘�1Þ, compute a shortest path pait0 . Denote

Ua�it0 ¼ haCpa

it0ðh‘�1Þ.

1.2. (Shortest hyperpath computation on Gb): For every x 2 W b, based on the current cost cbðf ‘�1Þ, compute

a shortest hyperpath pbx. Denote Ub�x ¼ �Cpb�x

ðh‘�1Þ.1.3. (Shortest combined hyperpath computation on G): For every x 2 W , compute

Uc�x;t0 ¼

1

cxUa�

it0 þ Ub�t0j ; for all t0 2 Tx:

1.4. For all x 2 W compute

kkx ¼ Uk�x þ lnðgk;‘�1

x Þ þ ak

b1

; k 2 fa; bg;

kcx ¼ mint02Tx

Uc�x;t0

(þlnðgc;‘�1

x;t0 Þ þ act0

b2

)þ lnðgc;‘�1

x Þ þ ac

b1

� lnðgc;‘�1x Þb2

: ð49Þ

Denote as t� the transfer node that minimizes the expression (49). Denote as pcx the shortest path on

the combined multi-modal network for the pair x through transfer node t�.1.5. Compute the solution �h‘ as

kx ¼ argmink2fa;b;cgfkkxg for all x 2 W ; �h‘pkxx

¼ �gx;

and set the others components to zero.


X‘x ¼ hx

8<: ¼ ðhpÞp2Q‘

x

Xp2Q‘

x

hp

�� ¼ �gx; hp P 0; for all p 2 Q‘x

9=;:

We propose to use a symmetric linearization method (see Wu et al., 1994; Montero and Bar-cel�o, 1996) to solve RMPVIPðX‘;C � KÞ. This method generates a sequence of nonlinear opti-mization problems in the following way. We denote as s the interior counter of this sequence ofnonlinear optimization problems. Let hs be a feasible solution for the RMPVIPð‘Þ, that is hs 2 X‘,then the nonlinear mapping is approximated by a symmetric linear mapping

CsðhÞ ¼ CðhsÞ þ 1

aBðhsÞTðh� hsÞ; ð40Þ

where BðhsÞ is the diagonal of the Jacobian matrix of C evaluated at hs (linearized Jacobi method,LJM for short) or BðhsÞ is some fixed symmetric and positive definite matrix B (projection method,PM in short). In both cases the RMVIPðX‘;C � KÞ is approximated at the point hs byRMVTPðX‘;C

s � KÞ and this symmetric VIP can be formulated as an optimization problem½OP‘ðsÞ�:


minCðhsÞTðh� hsÞ þ 1

2aðh� hsÞTBðhsÞðh� hsÞ þ RðgÞ; ð41Þ

subject to:

h 2 X‘; ð42Þg ¼ dgh; ð43Þ

where

RðgÞ ¼Xx2W

RxðgxÞ

¼Xx2W

"� ð1=b2Þgcxðln gcx � 1Þ þ ð1=b1Þ

Xk2fa;b;cg

gkxðln gkx � 1þ akÞ

þ ð1=b2ÞXt2Tx

gcx;tðln gcx;t � 1þ act Þ#

and X‘ is defined as (39). If BðhsÞ is a diagonal matrix, this program decomposes into a collectionof smaller and independent convex programs, one per origin–destination pair x, ½OP‘

xðsÞ�:

minXp2Q‘

x

CpðhsÞðhp�

� hsÞ þ 1

2aBpðhsÞðhp � hsÞ2

�þ RxðgxÞ;

subject to:

hx 2 X‘x; gx ¼ dgxhx:

where dgx is the demand/path flows incidence sub matrix associated with the O–D pair x. Thissubproblem can be solved using a type of Evans’ algorithm (Evans, 1976), where the shortesthyperpaths computation are easily solved.

3.2. Analysis of the convergence

Patriksson (1999) guarantee the convergence (Theorem 9.16) of the previous algorithms underthe hypothesis that the extended cost mapping CðhÞ � KðhÞ is monotone, Lipschitz continuous,and is C1 on X and under an exact solution of RMPVIPð�‘2 ¼ 0Þ.

Garc�ıa (2001) proves that if the cost mapping CðhÞ is monotone, Lipschitz continuous, and is

C1 on X thus the extended cost mapping CðhÞ � KðhÞ also holds these properties. It shows that

only the above assumptions on the cost function CðhÞ, and not for the extended cost mappingCðhÞ � KðhÞ, is required for the convergence of the simplicial decomposition algorithm.

On the other hand, the convergence of the linearization method for RMPVIPðX;C � KÞ can beensured under the assumptions of strongly monotone and Lipschitz continuous on CðhÞ, BðhsÞdoes not vary ‘‘too much’’ throughout the iterative process, and for certain values of a. If theprojection method is used the second condition is automatically satisfied, and the third one whichis stated as a is sufficiently small. This leads us to require the strongly monotone assumption forthe cost mapping to assure the global convergence of the simplicial decomposition method.


4. Experimental results

In this section, we effect several computational tests to research the behavior of the algorithmproposed in Section 3. The CG/SD algorithm was implemented in an algebraic modeling system,GAMS (an introduction to this computer environment is given in Castillo et al., 2002). Thecomputational experiments have been completed in Appendix A, where the use of the proposedmodel in multi-modal network design problems is illustrated.

4.1. Description of the test problems

The authors do not know real network models of MAPCM in literature. For this reason, twotest problems have been developed. The first problem is called Problem GM and the asymmetriccosts are derived from the public transport system. The second problem, Problem LH, consists of atraffic network with asymmetric transport costs and a public transport system with constant costs.

Symmetric problems for computational experiments have been also developed, which may beobtained in our reference Garc�ıa et al. (submitted for publication). In this paper we have preferredto concentrate on the aforementioned non-symmetric networks.

Problem GM simulates a multi-modal network of a corridor of a city. This city consists of twosuburban areas, each one modeled by two centroids, and the downtown area. It is assumed thatduring the period analyzed the commuters travel from the suburban area to the downtown area.Problem GM consists of a traffic network with separable costs and a public transportation systemmodeled by the passenger assignment model given in Cea and Fern�andez (1993). In this model thepassengers travel by following a sequence of intermediate transfer nodes, the so called route

sections. The passengers could bundle together a subset of available lines in order to reduce thewaiting time and hence the overall transit time for each route section. The determination of the setof ‘‘attractive’’ lines for the route sections is known as the common lines problem.

It is assumed that the congestion phenomena on a transit network is concentrated at transitstops. There, passengers experience waiting times that depend on the total capacity of theattractive set of lines considered and on the total number of passengers using that set of lines.When congestion exists a fraction of the vehicles arriving at a node i will be full. In order to tacklethis phenomena, the concept of effective frequencies was introduced by these authors instead ofnominal frequencies.

This model is formulated by the following variational inequality problem.[VIP(Xb, cb)]: Find ðf �; f �

L Þ 2 Xb so that

cbðf �; f �L Þ

Tðf � f �ÞP 0; for all ðf �; f �L Þ 2 Xb;

where ½cbðf �; f �L Þ� is a general volume-delay function and Xb is the space of feasible link flows,

which is defined by the following set of equations:

gbx ¼Xp2Pb

x

hp; 8w 2 W b [ Wtj; ð44Þ

f s ¼Xp2Pb

dsphp; 8s 2 B; ð45Þ

Table

Basic

L

LI

L2

L3

L4

L5


f sL ¼ p0s

LPL2Bs

p0sL

f s; 8L 2 Bs; 8s 2 B; ð46Þ

p0sL ¼ aL

aLpLþ bL

~fiLpLkL

� �nL ; 8L 2 Bs; 8s 2 B; ð47Þ

hp P 0; 8p 2 Pb; ð48Þ
being B is the set of route sections, Bs is the set of attractive lines for section s, pL is the nominalfrequency of the line L, p0s
L is the effective frequency of the line L in section s, kL is the capacity ofthe line L (number of places per vehicles), f s is the total number of passengers boarding the routesection s (route section flow), f s

L is the passengers traveling on line L over route section s (linesection flow), ~fiL is the number of passengers boarding line L before node i and alighting on someother node after it, aL; bL; n are calibration parameters.

Eq. (46) assign the passengers to bus lines proportionally to the effective frequencies of eachcommon line. Eq. (47) define a relationship between an equivalent average waiting time at thetransit stop i and the effective frequency of the line L at the stop i. Note that the computation of ahyperpath cost using the link flow f s requires the above system of nonlinear equations to besolved in order to obtain the effective frequencies for lines and route sections.

The parameters of Eqs. (46) and (47) can be observed in Table 3.The used transit network is represented in Fig. 1. The public transport system consists of five

transit lines serving the network. It is supposed that all the common lines have the same in-vehicletime travel time. As a result of this, all the lines are attractive to the traveler.

3

data for transit network of Problem GM

aL pL kL bL nL

1 13.49 10,000 1.65 4.00

1 11.27 10,000 1.37 4.00

1 23.95 20,000 2.94 4.00

1 15.33 20,000 1.88 4.00

1 33.61 30,000 4.13 4.00

Fig. 1. Transit network for Problem GM.

Fig. 2. Network topology for Problem GM.


The used volume-delay function for the route sections is

cbs ðf Þ ¼ �cs þ1

p0sðf Þ

; 8s 2 B;

where �cs is the in-vehicle travel cost (including the in vehicle travel time plus fare) and 1p0sðf Þ

is thewaiting time for route sections. To compute p0

sðf Þ it is necessary to solve the system of nonlinearequations (46) and (47).

The traffic network has 12 nodes and 22 links. The travel costs are all separable and theirgeneral form is

cal ðf Þ ¼ �cl þ 0:5ðfl=klÞ4; 8l 2 A;

where �cl is the free-flow travel component, and kl is the practical capacity of link l.Fig. 2 shows the topology of the multi-modal network and Table 4 the set of parameters for the

link costs and the demand model.Problem LH consists of asymmetric traffic network taken from Lawphongpanich and Hearn

(1984) and a public transport system which costs are independent of the link flows. The multi-modal network topology is shown in Fig. 3; and the trip matrix, parameters of the cost functionsand the nested logit model are shown in Table 5. This network has 56 links, 2 transfer nodes and24 nodes.

4.2. Computational testing

It is well known that, in the (restricted) simplicial decomposition algorithm for large-scaleoptimization problems, the vast majority (around 80–90% according to Hearn et al., 1987) of the

Table 4

Network parameters for Problem GM

Arc l O D �cl or �cs kl Arc l O D �cl or �cs kl

a1 1 5 0.38125 – a2 1 2 0.05475 –

a3 2 5 0.33625 – a4 2 3 0.11437 –

a5 3 5 0.23175 – a6 3 4 0.05288 –

a7 4 5 0.18888 – a8 5 10 0.00000 10,000

a9 6 1 0.00000 10,000 a10 6 11 0.00000 20,000

a11 7 12 0.00000 10,000 a12 7 2 0.00000 10,000

a13 8 3 0.00000 10,000 a14 8 13 0.00000 10,000

a15 9 4 0.00000 10,000 a16 9 14 0.00000 10,000

a17 10 5 0.00000 10,000 a18 11 12 0.05240 10,000

a19 11 15 0.05260 20,000 a20 12 15 0.04880 10,000

a21 12 13 0.16700 10,000 a22 12 2 0.16667 10,000

a23 13 3 0.16667 10,000 a24 13 14 0.06860 10,000

a25 13 16 0.08000 10,000 a26 14 4 0.16667 10,000

a27 14 10 0.28620 10,000 a28 14 17 0.07940 10,000

a29 15 16 0.20200 20,000 a30 15 12 0.04860 10,000

a31 16 13 0.07940 10,000 a32 16 17 0.09220 20,000

a33 17 14 0.07940 10,000 a34 17 10 0.26440 20,000

a35 11 1 0.16667 10,000

Demand model

O–D

pair

Travel

demand

Car

occupancy

rate

Logit nested model

(6, 10) 10,000 1.0 aa ¼ 0:5, b1 ¼ 5:0(7, 10) 20,000 1.0 ab ¼ 1:5, b2 ¼ 4:5(8, 10) 5,000 1.0 ac ¼ 1:0, ha ¼ 1:0(9, 10) 15,000 1.0 act ¼ 1:0, hb ¼ 1:0


computational effort is spent in the column generation phase. In the disaggregate version thesepercentages are essentially reversed. This is due to the difference of sizes of the master problems.The number of variables of the master problems is (assuming non-dropping of the columns) thenumber of generated routes for the disaggregated version but it is upper bounded by the numberof main iterations for the aggregated simplicial decomposition method.

We believe that this situation is also maintained in the context of variational inequalityproblems and it motivates us to study the possibilities of solving the RMPVIPs. Any convergentmethod may be applied to the master problem. In our testing we used the linearized Jacobimethod (LJM) and the projection method (PM) to approximate the RMPVIPs. In the projectionmethod the value of the parameter a was 10,000 for Problem GM and a ¼ 0:5 for Problem LH, inLJM the value used was a ¼ 1. The master problem need only be solved approximately, and theaccuracy of its solution is implicitly defined by the algorithm applied (LJM or PM) and by thenumber of OPs used.

This experiment has the objective of analyzing the accuracy of the RMPVIPs in the overallcomputational burden of the CG/SD algorithm. This quantity is the sum of the computationalefforts spent in CGPVIPs and of the one spent in RMPVIPs. The first addition is approximately

Fig. 3. Network topology for Problem LH.


linear on the number of main iterations and the second one depends on the total number of OPsused to approximate the RMVIPs.

Our implementation of CG/SD algorithm begins by using the routes of maximum cost as aninitial feasible solution and terminates when the value of the relative gap, defined to be

½Cðh‘�1Þ � Kðh‘�1Þ�Tðh‘�1 � �h‘Þ½�Cðh‘�1Þ � Kðh‘�1Þ�T�h‘

is less than 10�4.Figs. 4 and 5 show the computational results obtained for Problem GM and Problem LH

respectively. The total amount of main iterations versus the number of OPs solved per mainiteration is shown on the left of the figures, and the total amount of inner iterations versus thenumber of OPs solved per iteration, that is, the total amount of OPs for solving the RMPVIPs, isshown on the right of the figures.

The conclusion of this experiment is that if the accuracy of the solutions of RMPVIPs isaugmented by means of increasing the number of OPs per iteration, so the total number ofCGPVIPs and RMPVIPs is reduced. On the other hand, the computational cost in RMVIPs isincreased or decreased in function of the trade-off between a reduction in the number of mainiterations and an increase of the inner iterations (that is, the number of OPs per iteration) toobtain the approximate solution of RMPVIP.

A small number of OPs, between 1 and 5, reduces the overall computational effort by the testproblems, but if this value is too large, the computational cost begins to grow, because the

Table 5

Network parameters for Problem LH

Arc l O D clðf Þ Arc l O D clðf Þ

a1 1 9 T ðf1 þ f14Þ a2 1 12 T ðf2 þ 0:50f11 þ 0:25f12 þ 0:25f14Þ

a3 2 10 T ðf3 þ 0:25f10 þ f21 þ 0:25f22Þ a4 2 16 T ðf4 þ f10Þ

a5 3 19 T ðf5 þ 0:25f32 þ 0:25f33 þ f34Þ a6 3 22 T ðf6 þ f32Þ


a9 9 6 T ðf9 þ f22Þ a10 9 16 T ð0:50f3 þ 0:25f4 þ f10 þ 0:25f22Þ

a11 10 5 T ð0:25f2 þ f11 þ f13 þ 0:25f14Þ a12 10 12 T ðf2 þ f12Þ

a13 11 5 T ðf11 þ f13Þ a14 11 9 T ðf1 þ 0:25f2 þ 0:25f11 þ f14Þ


a17 13 15 T ðf17 þ f30Þ a18 13 20 T ðf18 þ 0:25f23 þ f24 þ 0:50f30Þ


a21 15 10 T ðf3 þ f21Þ a22 15 6 T ð0:25f3 þ 0:25f10 þ f22Þ

a23 16 14 T ðf17 þ 0:25f18 þ f23 þ f29 þ 0:25f30Þ a24 16 20 T ðf18 þ f24Þ

a25 17 11 T ðf19 þ f25Þ a26 17 13 ðT15 þ 0:25T16 þ 0:25T19 þ f20 þ f26Þ


a29 19 14 T ðf23 þ f29Þ a30 19 15 T ð0:50f17 þ 0:25f18 þ 0:25f23 þ f30Þ

a31 20 7 T ðf31 þ f33Þ a32 20 22 T ð0:25f5 þ f6 þ f32 þ 0:25f33Þ

a33 21 7 T ð0:25f5 þ 0:25f31 þ 0:50f32 þ f33Þ a34 21 19 T ðf5 þ f34Þ

a35 22 8 T ðf27 þ f35Þ a36 22 17 T ðf7 þ 0:25f8 þ 0:25f27 þ f36ÞT ðzÞ ¼ 3:0ð1þ 0:15ðz=35Þ4Þ

Public transport network

Arc l O D clðflÞ Arc l O D clðflÞ Arc l O D clðflÞ

a37 1 6 10 a38 1 7 10 a39 1 8 10

a40 2 5 10 a41 2 7 10 a42 2 8 10

a43 3 5 10 a44 3 6 10 a45 3 8 10

a46 4 5 10 a47 4 6 10 a48 4 8 10

a49 13 130 0:5þ 0:45ðf49=100Þ4 a50 14 140 0:5þ 0:45ðf50=100Þ4 a51 130 5 10

a52 130 6 10 a53 130 7 10 a54 130 8 10

a55 140 5 10 a56 140 6 10 a57 140 7 10

a58 140 8 10

Demand model

O–D pair Travel

demand

O–D Travel demand O–D

pair

Travel

demand

Car occu-

pancy rate

Logit nested model

(1, 6) 70 (1, 7) 60 (1, 8) 4 1.0 aa ¼ 1:0, b1 ¼ 0:05(2, 5) 80 (2, 7) 70 (2, 8) 50 1.0 ab ¼ 1:0, b2 ¼ 0:05(3, 5) 60 (3, 6) 40 (3, 8) 70 1.0 ac ¼ 2:0, ha ¼ 1:0(4, 5) 70 (4, 6) 50 (4, 7) 80 1.0 act ¼ 1:0, bb ¼ 1:0

246

R.Garc �ıa

,A.Mar �ın

/Transporta

tionResea

rchPart

B39(2005)223–254

Fig. 4. Computational results for Problem GM.

Fig. 5. Computational results for Problem LH.


computational costs for solving the RMPVIPs increases and it is not compensated for theeconomy of computational cost in CGPVIPs.

The next experiment has the objective of comparing the behavior of the PM and LJM. Theexperiment consists of computing the relative gap versus main iteration for a fixed value of OPs.

We used the value of 5 for Problem GM and 1 for Problem LH. The first value was motivatedby the fact that PM and LJM use the same amount of main iterations, and the second value is agood choice for the performance of the CG/SD algorithm.

The implementation of LJM requires a numerical approximation of the partial derivateoCp=ohp for Problem GM because the link cost on the route sections in the public transportsystem are implicitly known via the system of nonlinear equations (46) and (47).

We have used the value of CpðhÞ � CpðhÞ, where hp0 ¼ hp, 8p0 6¼ p and hp ¼ hp þ 1 as anapproximation of oCp=ohp. This implies that one system (46) and (47) per each partial derivate has

Fig. 6. Computing relative gap versus iteration.


to be solved in order to compute CpðhÞ and it becomes impractical even for networks of mediumsize. On the other hand, these partial derivates are explicitly computed for Problem LH, and thusLJM and PM spent the same CPU time to solve the same amount of OPs.

The computational results of this experiment are shown in Fig. 6. The conclusion is that thespeed of convergence of LJM is better than PM with respect to the number of main iterations.These results may indicate that LJM is preferable to PM when the partial derivates oCp=ohp ofMAPCM can be explicitly computed.

The CG/SD algorithm is convergent in this experiment. Nevertheless, the convergence may belost for extreme values of certain parameters of the network test. (We have experimented it onsome numerical tests in Appendix A.) This is due to the loss of the convergence of LJM and PMfor solving the RMPVIP. To avoid this drawback we have introduced the following modificationin LJM and PM. Let �hs be the current iterate in RMPVIP, let hsþ1 be the next iterate obtainedusing LJM or PM to approximate RMPVIP, thus the next modified iterate will be a convexcombination of the above iterate, that is �hsþ1 ¼ l�hs þ ð1� lÞhsþ1 where l 2 ð0; 1Þ. We have usedthe value of l ¼ 1=2 for the numerical tests of Appendix A.

5. Conclusions

The goal of this paper is to develop an approach to the multi-modal assignment problem withcombined modes to be used in the context of urban transport management models. In this way themain motivation is derived from our experience with the design of interconnecting high-qualitypublic–private systems, and especially the design of urban multi-modal interchanges.

In the paper a variational inequality problem to formulate the equilibrium multi-modalassignment problem with combined modes is presented, which may be used as a submodel of theabove network design problems. This model explicitly takes into account the choices of route,mode and transfer node, in a nested choice structure.

The model have great flexibility to formulate different assumptions because it does not assume agiven transit assignment model so it is possible to consider different modeling approaches. Thechosen approach may be used with fix and elastic demand and to deal with deterministic andstochastic traffic assignment models.


Our contributions in relation to the previous network equilibrium model with combined modes,Fern�andez et al. (1994), are the following: We consider the problem in the space of the hyperpathflows, which allows us to consider asymmetric costs. Great flexibility is given to the modeling ofthe traffic and transit networks. We have incorporated two important generalizations: elasticdemand and stochastic assignment. We have formulated the equilibrium conditions using avariational inequality formulation and we have developed a methodology to solve it.

The submodels must represent the reality as exactly as possible but they must be also solved asfar as possible, given that they will be called upon many times when the design alternatives of anetwork design problem is considered. In this way, we have chosen and developed a disaggregatedsimplicial decomposition algorithm for the variational inequalities formulation in the space of themulti-modal hyperpath flows.

The formulation in terms of the path flows is extremely operative, given that the paths aregenerated when they are required, and only the equilibrium ones are used.

This formulation and algorithm are very useful for reoptimizations propose, and this is veryimportant in the solving of submodels in network design problems. Furthermore, this algorithmhas excellent possibilities for the parallel computation implementation. We think that the resultingalgorithm is a computationally tractable way of solving large-scale multi-modal assignmentproblems. Moreover, due to the excellent restarting capabilities, this approach seems attractive forother asymmetric assignment models which may be formulated in the space of hyperpath/pathflows.

In this paper the influence of different algorithmic alternatives has been computationallystudied, and some model capacities have been illustrated in the context of the urban multi-modalinterchange design problem.

Appendix A. Illustration of the use of MAPCM

Improvement of multi-modal urban transport network characteristics will induce changes intraffic flow over the network. Therefore, predictions of traffic patterns via a user behavior modelare essential to the multi-modal network design process. In this subsection, we have numericallyillustrated some possibilities for the use of MAPCM as a users’ behavioral submodel of a con-tinuous or discrete network design problem.

MAPCM deals with three macro-effects: the congestion effect that produces redistribution offlows and demand among transport modes, the competition of parking lots, and the elasticity ofthe parking demand versus parking design parameters. These capabilities have been illustratedusing the following examples:

Example 1. Influence of the number of vehicles in a transit line on the modal split.

Example 2. Influence of the locations of the park’n ride stations on the park’n ride demand.

Example 3. Park’n ride demand versus parking capacity.

Example 4. Park’n ride demand versus parking fares.

Fig. 7. Modal split versus frequencies for transit Line 4.


Example 1: MAPCM can be used to evaluate the modal split of the demand (by modes and bytransfer nodes) versus the resource assignment in the public transport system in the context ofmulti-modal routes, demand characterization, transport network, generalized costs, and othermacro characteristics of the transport system.

The nominal frequency of a transit line depends linearly on its transit vehicles. This relationshipis used to introduce the modifications in the number of vehicles which are assigned to a specifictransit line in MAPCM. The modal split of the demand in function of the number of vehicles forthe transit Line 4 is estimated for Problem GM in this example. Fig. 7 shows the results obtained.It is observed that an increase of the number of transit vehicles reduces the share market of privatetransportation, and increases the share market of public transportation. This result may bethought to be a priori, but using MAPCM this effect may be quantified.

Example 2: MAPCM can be used to evaluate the installation of new parking lots. This eval-uation takes into account those already existing and the influences between all parking lots. Toillustrate this use of the model, we have dealt with Problem GMwhich has four available locationsfor the parking lots. They are defined by the links: a35, a22, a23 and a26. In this experimentMAPCM is solved for all possible configurations and the park’n ride demand is reported via eachparking lot. The results obtained are shown in Table 6. For example if we install only two parkinglots at links a35 and a26 the number of trips using the park’n ride mode could be 5523.68, fromwhich 1094.55 users transfer via a parking lot located at link a35 and 4429.14 users via parking lota26. Table 6 shows that the level of service of the parking lots depends on the competence betweenparking lots. In this example the fact that the installation of a new parking lot is an element ofgeneration of combined trips is also illustrated.

Example 2 may be a case of a user behavior submodel for a discrete network design problemwhich is concerned with the addition or removal of the parking links in function of a trade-offbetween user benefits and the cost of the network alteration.

Examples 3 and 4: The enhancement of parking capacity can be seen as an incentive rule inorder to promote the use of combined trips. On the other hand, the pricing of parking lots is a toolthat reduces the incentive to use combined trips. Examples 3 and 4 illustrate how the para-

Table 6

Park’n ride demand versus location of parking lots

Location Combined mode trips

a35 a22 a23 a26 Total

a35 a22 a23 a26 1003.74 3237.96 2549.03 3360.61 10,151.34

– a22 a23 a26 – 3054.39 2859.41 3861.03 9774.83

a35 – a23 a26 1000.25 – 3148.33 4082.54 8231.13

a35 a22 – a26 1046.93 3768.41 – 3850.92 8666.26

a35 a22 a23 – 1018.63 3416.66 2720.00 – 7155.29

– a23 a26 – – 3129.77 4129.42 7259.18

– a22 – a26 – 3506.76 – 4258.33 7765.09

– a22 a23 – 3272.02 3058.35 6330.36

a35 – – a26 1094.55 – – 4429.14 5523.68

a35 – a23 – 1013.98 – 3281.95 – 4295.93

a35 a22 – – 1124.42 3703.78 – – 4828.20

– – – a26 – – – 3797.96 3797.96

– – a23 – – – 3321.10 – 3321.10

– a22 – – – 3727.53 – – 3727.53

a35 – – – 1248.62 – – – 1248.62


meterization of the parking links may take into account both policies. These examples take thenetwork topology as given and are concerned with the parking link parameters.

Examples 3 and 4 are defined using the LH network which has two parking lots located at thelinks a49 and a50. Two demand models are used, the first one is defined by the parametersb1 ¼ b2 ¼ 0:05 and this choice simplifies the nested logit model to a multinomial logit modelwhere the park’n ride alternative per each distinct transfer node is considered as a distinct modalalternative for O–D pairs; the second one is denned letting b1 ¼ 0:05 < b ¼ 0:1 which is aproperly nested logit model. To increase the level of congestion we have used the link costclðflÞ ¼ 50 for the links a37 to a48 in the public transport system.

Experiment 3 consists of changing the fare of parking lot a49 and evaluating the level of serviceof the parking lots a49 and a50 using the two above demand models. This parking pricing is carriedout by increasing the free-flow cost parameter and thus, the fare is expressed in time cost.

Fig. 8. Park’n ride demand versus parking fares.

Fig. 9. Park’n ride demand versus parking capacity.


The results of Experiment 3 are shown in Fig. 8. It is shown that an increase of the fares inparking lot a49 reduces the amount of passengers in this parking lot, and the users are transferredto another mode of transportation. A few of these users are routed through parking lot a50.

Experiment 4 evaluates the parking demand in function of the capacity of parking lot a49 usingthe above two demand models. The results of Experiment 3 are shown in Fig. 9. An increase of thecapacity of the parking lot generates an increase in the parking demand when the capacity of theparking lot is not enough. When the supply of parking is greater than a level, an increase ofthe capacity of parking does not generate an increase in the demand. Note that an improvementin the facilities in parking lot a49 slightly reduces the level of service of the competence of parkinglot a50.

Experiments 3 and 4 shows that the nested logit model is more sensible than the multinomiallogit model to the interactions between parking lots. This may indicate that the nested logit modelis more adequate in capturing some correlations among parking alternatives.

References

Ben-Akiva, M.E., Bowman, J.L., 1998. Activity based travel demand model systems. In: Marcotte, P., Nguyen, S.

(Eds.), Advanced Transportation Modelling. Kluwer Academic Publishers, Massachusetts, pp. 27–46.

Bertsekas, D.P., Gafni, E., 1982. Projections methods for variational inequalities with applications to the traffic

assignment problem. Mathematical Programming Study 17, 139–159.

Bifulco, G.N., 1993. A stochastic user equilibrium assignment model for la evaluation of parking policies. European

Journal of Operational Research 71, 269–287.

Bouza€ıene-Ayari, B., Gendreau, M., Nguyen, A., 1998. Passenger assignment in congested transit networks: a historical

perspective. In: Marcotte, P., Nguyen, S. (Eds.), Advanced Transportation Modelling. Kluwer Academic Publishers,

Massachusetts, pp. 47–71.

Boyce, D.E., 1984. Urban transportation network-equilibrium and design models: recent achievements and future

prospects. Environment and Planning A 16, 1445–1474.

Cantarella, G.E., 1997. A general fixed-point approach to multimode multi-user equilibrium assignment with elastic

demand. Transportation Science 31, 107–128.

Castillo, E., Conejo, A., Pedregal, P., Garc�ıa, R., Alguacil, N., 2002. Building and Solving Mathematical Programming

Models in Engineering and Science. In: Pure and Applied Series (PAMS). John Wiley & Sons, New York.


Cea, J.D., Fern�andez, E., 1993. Transit assignment for congested public transport systems: an equilibrium model.

Transportation Science 27, 133–147.

Erlander, S., 1977. Accessibility, entropy and the distribution and assignment of traffic. Transportation Research 11,

149–153.

Evans, S.P., 1976. Derivation and analysis of some models for combining trip distribution and assignment.

Transportation Research 10, 37–57.

Fern�andez, E., Friesz, T.L., 1983. Equilibrium predictions in transportations markets: the state of the art.


Fern�andez, E., Cea, J.D., Florian, M., Cabrera, E., 1994. Network equilibrium models with combined modes.


Ferrari, P., 1999. A model of urban transport management. Transportation Research B 33, 43–61.

Florian, M., 1977. A traffic equilibrium model of travel by car and public transit modes. Transportation Science 11,

166–179.

Florian, M., Los, H., 1978. Determining intermediate origin–destination matrices for the analysis of composite mode

trips. Transportation Research B 13, 91–103.

Florian, M., Spiess, H., 1983. On binary mode choice/assignment models. Transportation Science 17, 32–47.

Florian, M., Nguyen, S., Ferland, 1977. On the combined distribution-assignment of traffic. Transportation Science 9,

43–53.

Garc�ıa, R., 2001. Metodolog�ıa para el dise~no de redes de transporte y para la elaboraci�on de algoritmos en

programaci�on matem�atica convexa diferenciable. Ph.D. Thesis, Escuela T�ecnica Superior de Ingenieros Aerona�uti-cos, Universidad Polit�ecnica de Madrid.

Garc�ıa, R., Mar�ın, A., 2001. Urban multimodal interchange design methodology. In: Pursula, M., Niittym€aki, J. (Eds.),Mathematical Methods on Optimization in Transportation Systems. In: Applied Optimization, vol. 48. Kluwer

Academic Publishers, Dordrecht, pp. 49–79.

Garc�ıa, R., Mar�ın, A., 2002. Parking capacity and pricing in park’n ride trips: a continuous equilibrium network design

problem. Annals of Operations Research 116, 153–178.

Garc�ıa, R., Mar�ın, A., Patriksson, M., 2003. Column generation algorithms for nonlinear optimization, I: Convergence

analysis. Optimization 52 (2), 171–200.

Garc�ıa, R., Mar�ın, A., Patriksson, M., submitted for publication. Column generation algorithms for nonlinear

optimization, II: Experimental study.

Hearn, D.W., Lawphongpanich, S., Ventura, J.A., 1987. Restricted simplicial decomposition: computation and

extensions. Mathematical Programming Study 31, 99–118.

Hunt, J.D., Teply, S., 1993. A nested logit model of parking location choice. Transportation Research B 127, 253–

265.

Lam, W.H.K., Huang, H.-J., 1992. A combined trip distribution and assignment model for multiple user classes.


Larsson, T., Patriksson, M., 1992. Simplicial decomposition with disaggregated representation for the traffic

assignment problem. Transportation Science 26, 4–17.

Larsson, T., Patriksson, M., Rydergren, C., 1997. Applications of simplicial decomposition with nonlinear column

generation to nonlinear network flows. In: Pardalos, P.M., Hager, W.W., Hearn, D.W. (Eds.), Network

Optimization. In: Lecture Notes in Economics and Mathematical Systems, No. 450. Springer-Verlag, Berlin, pp.

346–373.

Lawphongpanich, S., Hearn, D.W., 1984. Simplicial decomposition of the asymmetric traffic problem. Transportation

Research B 18, 123–133.

LeBlanc, L.J., Farhangiam, K., 1981. Efficient algorithms for solving elastic demand traffic assignment problems and

mode split-assignment problems. Transportation Science 15, 306–317.

Ludgren, J.T., Patriksson, M., 1997. An algorithm for the combined distribution and assignment model. Technical

report, Department of Mathematics, Link€oping Institute of Technology, Linkoping Institute of Technology,

Link€oping, Sweden.Marcotte, P., Gu�elat, J., 1988. Adaptation of a modified Newton method for solving the asymmetric traffic equilibrium

problem. Transportation Science 22, 112–124.


Montero, L., Barcel�o, J., 1996. A simplicial decomposition algorithm for solving the variational inequality formulation

of the general traffic assignment problem. TOP, Journey of the Spanish Statistical and Operation Research Society 4,

225–256.

Nagurney, A.B., 1984. Comparative tests of multimodal traffic equilibrium methods. Transportation Research B 18,

469–485.

Nguyen, S., Pallotino, S., 1988. Equilibrium traffic assignment for large scale transit network. European Journal of

Operational Research 37, 176–186.

Pang, J.S., Yu, C.S., 1984. Linearized simplicial decomposition methods for computing traffic equilibria on networks.

Networks 14, 427–432.

Patriksson, M., 1994. Traffic Assignment Problem. Models and Methods. VSP, Utrecht, The Netherlands.

Patriksson, M., 1999. Nonlinear Programming and Variational Inequality Problems. A Unified Approach. Kluwer

Academic Publishers, Dordrecht.

Spiess, H., Florian, M., 1989. Optimal strategies: a new assignment model for transit networks. Transportation

Research B 23, 83–102.

Toint, P., Wynter, L., 1995. Asymmetric multiclass traffic assignment: a coherent formulation. In: Proceedings of

International Symposium on Transportation and Traffic Theory.

Wong, S.C., 1998. Multicommodity traffic assignment by continuum approximation of network flow with variable

demand. Transportation Research B 32 (8), 567–581.

Wu, J.H., Florian, M., 1993. A simplicial decomposition method for the transit equilibrium assignment problem.

Annals of Operations Research 44, 245–260.

Wu, J.H., Florian, M., Marcotte, P., 1994. Transit equilibrium assignment: a model and solution algorithms.


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Network equilibrium with combined modes: models and solution algorithms

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